If such a locally unbiased estimator exists, the Cramér-Rao bound can always be saturated. However, the identification of the saturation conditions of the Cramér-Rao bound is a complex subject that involves a variety of technical details. In particular, for the probability distribution satisfying equation (REF ), there exists a local unbiased estimator \(\hat{ \theta }\)  saturating the Cramér-Rao bound provided that [1]}
\(\frac{\partial \log p\left(x; \theta \right)}{\partial \theta }=\mathbb {F}\left( \theta  \right)\left(\hat{ \theta }- \theta \right).\) 
